cygv

This project implements an efficient algorithm to perform the HKTY procedure [1, 2] to compute Gopakumar-Vafa (GV) and Gromov-Witten (GW) invariants of Calabi-Yau (CY) manifolds obtained as hypersurfaces or complete intersections in toric varieties. This project is based on the work presented in the paper "Computational Mirror Symmetry", but written in the Rust programming language and with some additional improvements.

Python package

The cygv package on PyPI ships prebuilt wheels, so no Rust toolchain is needed to install it.

pip install cygv

It exposes two functions, compute_gv and compute_gw, which take the data describing the CY and return the corresponding invariants. Every vector argument is given as a list of vectors, or as any array-like that numpy accepts.

from cygv import compute_gv, compute_gw

# The generators of the Mori cone.
generators = [[0, -1], [1, 2]]
# A vector that has a positive inner product with every generator.
grading_vector = [3, -1]
# The GLSM charge matrix, given as one charge vector per Kähler parameter.
q = [[1, 1, 1, 0, 1, 2], [0, 0, -1, 1, 1, -1]]
# The triple intersection numbers. Only one ordering of each triplet may be given.
intnums = {(0, 0, 0): 2, (0, 0, 1): 1, (0, 1, 1): -1, (1, 1, 1): 5}

compute_gv(generators, grading_vector, q, intnums, max_deg=3)
# [((0, -1), -4), ((1, 2), 252), ((1, 1), 7524), ((1, 0), 7524), ...]

compute_gw(generators, grading_vector, q, intnums, max_deg=3)
# [((0, -1), Fraction(-4, 1)), ((0, -2), Fraction(-1, 2)), ((1, 2), Fraction(252, 1)), ...]

How many curve classes are computed is controlled by max_deg, min_points, or target_points, of which at most one may be given. When none of them is given, the generators are taken to be the complete list of curve classes to use. Complete intersections are specified by passing the nef partition as nefpart. The computation is done with exact rational arithmetic by default; passing a number of bits as prec switches it to floating-point arithmetic, which is much faster for large computations. Rational arithmetic is nevertheless the recommended choice, as there is no way to know beforehand how much precision a given computation needs: too low a prec makes the GV computation fail, and makes the GW one silently return wrong invariants.

The results are returned as a list of (curve_class, invariant) pairs, in no particular order. GV invariants are Python ints, and GW invariants are fractions.Fractions, or mpmath.mpfs when prec is given. For CYs of dimension greater than three the invariants are labeled by a curve class and by the index of a reference surface, which is the first index of the intersection numbers, so each entry looks like (((0, -2, 0, 1, 0, 2), 3), -24) instead.

The computation runs in a subprocess, so it can be interrupted with ctrl+c without losing the Python session.

Command line interface

This project also ships a cygv executable, so that it can be used without Python. It reads YAML-formatted data from a file, or from the standard input, and writes the resulting invariants to a file, or to the standard output.

cargo install cygv
cygv --file input.yaml --output invariants.yaml

It is part of the default cli feature. Library users that do not need it can depend on this crate with default-features = false to avoid pulling in the argument and YAML parsers.

The input is a stream of YAML documents, each of which specifies a single CY manifold.

---
name: an example threefold
generators: [[0, -1], [1, 2]]
grading_vector: [3, -1]
q: [[1, 1, 1, 0, 1, 2], [0, 0, -1, 1, 1, -1]]
intnums:
  - [0, 0, 0, 2]
  - [0, 0, 1, 1]
  - [0, 1, 1, -1]
  - [1, 1, 1, 5]
invariants: gv
min_points: 100

The intersection numbers can equivalently be given as a mapping from indices to values, which is easier to read when there are many of them.

intnums:
  "0,0,0": 2
  "0,0,1": 1
  "0,1,1": -1
  "1,1,1": 5

The results are written back as one YAML document per input document, sorted by the degree of the curve classes under the grading vector.

---
name: an example threefold
invariants: gv
is_threefold: true
grading_vector: [3, -1]
results:
  - {curve_class: [0, -1], degree: 1, gv: -4}
  - {curve_class: [1, 0], degree: 3, gv: 7524}

Run cygv --help for the full list of input fields and options, and see the examples directory for complete inputs.

License

Licensed under the GNU General Public License v3 or later (GPLv3+) (LICENSE or https://www.gnu.org/licenses/gpl-3.0.en.html#license-text).

Contribution

Any contribution submitted for inclusion in the work by you, shall be licensed as under the GPLv3+, without any additional terms or conditions. See CONTRIBUTING.md for guidelines.